Polar & exponential form
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Flip to reveal answersWhat is the modulus r of z = a + bi?
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All 8 Flashcards — Polar & exponential form
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Question
What is the modulus r of z = a + bi?
Answer
r = |z| = √(a² + b²) — its distance from the origin on the Argand diagram.
Question
What is the argument of z?
Answer
The angle θ from the positive real axis (anticlockwise positive); fix the quadrant by sketching the point.
Question
Write the three equivalent polar/exponential forms of z.
Answer
z = r(cos θ + i sin θ) = r cis θ = r e^(iθ).
Question
How do you multiply two complex numbers in polar form?
Answer
Multiply the moduli and add the arguments: z₁z₂ = r₁r₂ e^(i(θ₁+θ₂)).
Question
How do you divide two complex numbers in polar form?
Answer
Divide the moduli and subtract the arguments: z₁/z₂ = (r₁/r₂) e^(i(θ₁−θ₂)).
Question
Convert z = 1 + √3 i to exponential form.
Answer
r = √(1+3) = 2, θ = arctan(√3) = π/3, so z = 2 e^(iπ/3).
Question
Convert 4 e^(iπ/6) to a + bi.
Answer
4(cos π/6 + i sin π/6) = 4(√3/2 + i/2) = 2√3 + 2i.
Question
Geometrically, what does multiplying by r e^(iθ) do to the Argand diagram?
Answer
It scales (stretches) by r and rotates by angle θ — that's why complex numbers model rotations and phase shifts.
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Full study notes for Polar & exponential form
Topic 1.13 hub
Complex numbers: continued (HL only)
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